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Simplifying (0.4 + -3a)(0.7 + -6a) = 0 Multiply (0.4 + -3a) * (0.7 + -6a) (0.4(0.7 + -6a) + -3a * (0.7 + -6a)) = 0 ((0.7 * 0.4 + -6a * 0.4) + -3a * (0.7 + -6a)) = 0 ((0.28 + -2.4a) + -3a * (0.7 + -6a)) = 0 (0.28 + -2.4a + (0.7 * -3a + -6a * -3a)) = 0 (0.28 + -2.4a + (-2.1a + 18a2)) = 0 Combine like terms: -2.4a + -2.1a = -4.5a (0.28 + -4.5a + 18a2) = 0 Solving 0.28 + -4.5a + 18a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.01555555556 + -0.25a + a2 = 0 Move the constant term to the right: Add '-0.01555555556' to each side of the equation. 0.01555555556 + -0.25a + -0.01555555556 + a2 = 0 + -0.01555555556 Reorder the terms: 0.01555555556 + -0.01555555556 + -0.25a + a2 = 0 + -0.01555555556 Combine like terms: 0.01555555556 + -0.01555555556 = 0.00000000000 0.00000000000 + -0.25a + a2 = 0 + -0.01555555556 -0.25a + a2 = 0 + -0.01555555556 Combine like terms: 0 + -0.01555555556 = -0.01555555556 -0.25a + a2 = -0.01555555556 The a term is -0.25a. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25a + 0.015625 + a2 = -0.01555555556 + 0.015625 Reorder the terms: 0.015625 + -0.25a + a2 = -0.01555555556 + 0.015625 Combine like terms: -0.01555555556 + 0.015625 = 0.00006944444 0.015625 + -0.25a + a2 = 0.00006944444 Factor a perfect square on the left side: (a + -0.125)(a + -0.125) = 0.00006944444 Calculate the square root of the right side: 0.008333333 Break this problem into two subproblems by setting (a + -0.125) equal to 0.008333333 and -0.008333333.Subproblem 1
a + -0.125 = 0.008333333 Simplifying a + -0.125 = 0.008333333 Reorder the terms: -0.125 + a = 0.008333333 Solving -0.125 + a = 0.008333333 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = 0.008333333 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = 0.008333333 + 0.125 a = 0.008333333 + 0.125 Combine like terms: 0.008333333 + 0.125 = 0.133333333 a = 0.133333333 Simplifying a = 0.133333333Subproblem 2
a + -0.125 = -0.008333333 Simplifying a + -0.125 = -0.008333333 Reorder the terms: -0.125 + a = -0.008333333 Solving -0.125 + a = -0.008333333 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = -0.008333333 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = -0.008333333 + 0.125 a = -0.008333333 + 0.125 Combine like terms: -0.008333333 + 0.125 = 0.116666667 a = 0.116666667 Simplifying a = 0.116666667Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.133333333, 0.116666667}
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